---
tag: 4NMTSX0XDS
kind: concept
name: Group ring
msc: [16S34, 20C05]
specializes: [1HM7ZYMPD7]
sources:
  - url: https://en.wikipedia.org/wiki/Group_ring
same_as:
  wikidata: Q2602722
  wikipedia: Group ring
---

## Summary

The group ring $R[G]$ of a [[group|G5S1M9]] $G$ over a [[ring|0B1261B5ED]] $R$ is the ring of finite formal sums $\sum_g a_g g$ with coefficients in $R$.
Such sums are multiplied using the group law.
It turns questions about $G$ and its actions into questions about modules over one ring.
When $R$ is a field, it is the [[group algebra|MNP2HBWB5Q]], the basic object of representation theory.
It is the special case of the [[monoid ring|1HM7ZYMPD7]] in which the monoid is a group.

## Explanations

### L1 human

Think of the elements of $G$ as basis vectors, and allow coefficients from $R$.
Multiply two formal sums by expanding and multiplying group elements in $G$.
For the integers $\Z$ and the cyclic group $\set{1, t}$ of order $2$, the element $1 - t$ satisfies $(1 - t)^2 = 1 - 2t + t^2 = 2 - 2t = 2(1 - t)$.
The integral group ring $\Z[G]$ is important in topology and in the study of group extensions.